Template:Zhegalkin twins 32

The pairs of matrices shown in the following boxes are twins. This is shorthand for the fact, that their rows are Zhegalkin twins. (Compare Zhegalkin matrix.)

The left side of each pair shows the truth tables of variadic logical connectives. (Compare these 16×16 matrices, which also show the arguments.)

Some of them are unusual, and have unusual names:
SAND (a.k.a. minimal negation operator) could be called all but one. Its reflection SNOR could be called no but one.
The name XOR is used for the parity function. The reflection of not XOR is called XAND. (Because the reflection of not OR is AND.)
GAND is SAND extended by AND. Its reflection GNOR is SNOR extended by not OR.
In ESAND and OSAND this extension happens only for an even or odd number of arguments. Their reflections are ESNOR and OSNOR.
EQ makes sense as generalization of the biconditional, and is true if all arguments have the same truth value (but not if there are no arguments).

The right side of each pair is a lower triangular matrix. Its entries are part of the Sierpinski triangle, which is the twin of not OR.

Each of these triangles is symmetric to another one (in two cases to itself). Only the triangle rows are symmetric (not the matrix rows).
Pairs with symmetric triangles are in the same box, e.g.:    XOR / OSAND,    SNOR / OSNOR

Some triangles are relative complements in the Sierpinski triangle:    TRUE / OR,    AND / EQ,    GNOR / SNOR,    OSNOR / ESNOR

TRUE		AND

not AND

not OR

OR		EQ

not XOR		ESAND

XOR		OSAND

not XAND		SAND

XAND		GAND

GNOR		ESNOR

SNOR		OSNOR

upper triangular
AND not OR These matrices were also in the last section. This section also shows pairs of matrices whose rows are Zhegalkin twins. The binary patterns are the same as in the lower triangular matrices in the section before, but they are horizontally and vertically flipped. Just like in the section before, there are pairs of matrices whose triangle rows are horizontally mirrored. Here they are marked with the same color. The labels in this section are to be understood like this: Take the lower triangular twin with that label, flip it horizontally and vertically, and make the twin of that. E.g. take the XOR twin from the last section. Flip it to get not XAND in this section. The twin of that is XOR in this section. (Half of these rows are noble, i.e. their own twins.)
OSNOR	OSAND	SAND	SNOR
not XAND	XOR	XAND	not XOR
ESNOR	ESAND	GAND	GNOR
OR	not AND	EQ